<Zone Plate>
A zone plate is an optical element called an annular zone plate of Fresnel, and is an optical element having a function of converging or diverging a wave using a diffraction phenomenon. As shown in FIG. 1 (a), when a point light source of wavelength λ is placed at a point Q away from the screen by a distance f on an optical axis 2, denoting a distance from an intersection O of the screen 5 and the optical axis 2 by r(x, y), the zone plate has a structure that divides the screen so that the distance from the point Q to the screen may become f+λ/2, f+λ, f+3λ/2, . . . f+mλ/2, i.e., a structure that is comprised of concentric annular zones whose center is the intersection O of the screen 5 and the optical axis 2 and that are different in width.
Generally, the zone plate that makes a light wave impossible to pass through the annular zones by blocking an odd-numbered annular zone to an adjacent eve-numbered annular zone when counting the degree n from the smaller side is called a positive zone plate, and the zone plate whose the even-numbered annular zone to the odd-number annular zone are blocked is called a negative zone plate. FIG. 1 (b) and FIG. 1 (c) show the positive and negative zone plates, respectively.
If the screen 5 of FIG. 1 (a) is replaced with a plane wave, the zone plate can be described as an interference fringe of the plane wave and a spherical wave. Formula 1 and Formula 2 express a plane wave Φp and a spherical wave Φs that propagate on the optical axis 2, respectively. However, it is assumed that both the plane wave and the spherical wave have an amplitude of 1 and theirs distributions are uniform. Handling about the amplitude is the same also in following formulae unless it is specified otherwise especially.
                                          Φ            p                    ⁡                      (                          x              ,              y                        )                          =                  exp          ⁡                      [                                                            2                  ⁢                  π                  ⁢                                                                          ⁢                  i                                λ                            ⁢              ηλ                        ]                                              [                  Formula          ⁢                                          ⁢          1                ]                                                      Φ            s                    ⁡                      (                          x              ,              y                        )                          =                  exp          [                                                    2                ⁢                π                ⁢                                                                  ⁢                i                            λ                        ⁢                                                            r                  ⁡                                      (                                          x                      ,                      y                                        )                                                  2                            f                                ]                                    [                  Formula          ⁢                                          ⁢          2                ]            
Here, η contained in the phase term of the plane wave is a phase value at the point O where observation is performed, and corresponds to an initial phase when forming an interference fringe. In this application, this will be called an initial phase of an annular zone grating that forms the zone plate, or is simply called an initial phase of the zone plate.
Moreover, an intensity distribution I(x, y) of the interference fringe that the above-mentioned plane wave Φp and spherical wave Φs make is expressed by Formula 3.
                              I          ⁡                      (                          x              ,              y                        )                          =                              1            2                    +                                    1              2                        ⁢                          cos              [                                                                    2                    ⁢                    π                                    λ                                ⁢                                  (                                                                                                              r                          ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                          2                                            f                                        -                                                                  3                        ⁢                        λ                                            4                                                        )                                            ]                                                          [                  Formula          ⁢                                          ⁢          3                ]            
Here, the distribution is expressed by setting the phase value of the plane wave to η=¾ in order to match the distribution to the positive zone plate and is expressed by normalizing the distribution so that a maximum value of the intensity may become 1. A value η=¾ of the initial phase of this plane wave is a condition that a diameter of a circular opening in a central part of the positive zone plate becomes a maximum. Moreover, the diameter has become just the same amount as that of the Zernike's phase plate, and this indicates a part of an effect as the phase plate of the zone plate.
When Formula 3 is binarized, the zone plate where an area of 0≦I<0.5 is made black (blocking) and an area of 0.5≦I<1 is made white (transmission) is the positive zone plate (FIG. 1 (c)); conversely, the zone plate where an area of 0≦I<0.5 is made white (transmission) and an area of 0.5≦I<1 is made black (blocking) is the negative zone plate (FIG. 1 (c)).
FIG. 2 (a) shows the intensity distribution I(x, y) based on Formula 3. In the central part of the figure, i.e., a portion of the center of the annular zone, there is a portion whose intensity is slightly weak, and this is an effect of the phase value η=¾ of the plane wave. FIG. 2 (b) shows an intensity distribution of the interference fringe at the time of η=0. Although at this time, the intensity becomes strongest in the central part, diameters of the circular opening in the central part (innermost annular zone) and respective annular zones have become small. That is, phases of the annular zones of which the zone plate is comprised change.
However, since a geometrical optical spatial relationship shown in FIG. 1 (a) does not changed at all with only a relative phase of the spherical wave and the plane wave that come to interfere with each other changed, the converging action and imaging action as the zone plate do not change between the both zone plates of FIGS. 2 (a), (b) in any way. FIG. 2 (c) shows the intensity distribution I(x, y) of the interference fringe at the time of η=½. Comparing this with FIG. 2 (b), the phase of the annular zone changes exactly by π or −π, and this distribution corresponds to the negative zone plate with respect to a condition of η=0.
Hereinafter, in this application, because of convenience of explanation, both a pattern that has an intensity distribution in the continuous half-tone as an interference fringe shown in FIG. 2 and a pattern that has a binarized intensity distribution as shown in FIG. 1 are called the zone plate unless it is specified otherwise especially. Further, if needed, the zone plate shall be differentiated by calling it the zone plate having the binarized intensity distribution. Moreover, although the above-mentioned description was explained taking the light wave as an example, such a relationship holds in a general wave of an X-ray, an electron ray, etc. and it is not limited to the light wave. This will be the same also in the following explanation.
<Imaging Action of Zone Plate>
Since the zone plate is described by interference of the spherical wave from the point light source and the plane wave, the zone plate may be considered as a hologram of the point light source. That is, the pattern expressed by Formula 3 has the imaging action like the hologram. Since it is the hologram, at the same time when a real image is formed, a virtual image (conjugate image) is also formed.
This can be briefly explained if the intensity distribution I(x, y) of the interference fringe of Formula 3 is redescribed as an amplitude transmissivity. An area where the intensity distribution is large is shown in Formula 4 as an area where the amplitude transmissivity Ψt(x, y) is large. This means positive in the case of a film.
                                          Ψ            t                    ⁡                      (                          x              ,              y                        )                          =                                            k              t                        2                    +                                                    k                t                            2                        [                                                  ⁢                                          exp                [                                                                            2                      ⁢                      π                      ⁢                                                                                          ⁢                      i                                        λ                                    ⁢                                      (                                                                                                                        r                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                2                                                f                                            -                                                                        3                          ⁢                          λ                                                4                                                              )                                                  ]                            +                              exp                [                                                                                                    -                        2                                            ⁢                      π                      ⁢                                                                                          ⁢                      i                                        λ                                    ⁢                                      (                                                                                                                        r                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                2                                                f                                            -                                                                        3                          ⁢                          λ                                                4                                                              )                                                  ]                                      ]                                              [                  Formula          ⁢                                          ⁢          4                ]            
In order to make the following discussions intelligible, Formula 4 was expressed by using an exponential function, but if penetrability kt of the entire plate is uniform and can be assumed to be kt=1, the right hand side coincides with the intensity distribution of the interference fringe that is given by Formula 3 and is expressed with a cosine function. When the plane wave of wavelength λ′ and initial phase η enters into the zone plate with the amplitude transmissivity expressed by Formula 4, the wave immediately after transmission of the zone plate is expressed by Formula 5.
                                          Φ            re                    ⁡                      (                          x              ,              y                        )                          =                                            1              2                        ⁢                          exp              ⁡                              [                                                                            2                      ⁢                      π                      ⁢                                                                                          ⁢                      i                                                              λ                      ′                                                        ⁢                                      ηλ                    ′                                                  ]                                              +                                    1              2                        [                                          exp                [                                                                            2                      ⁢                      π                      ⁢                                                                                          ⁢                      i                                                              λ                      ′                                                        ⁢                                      (                                                                                                                        r                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                2                                                                                                      λ                                                          λ                              ′                                                                                ⁢                          f                                                                    -                                                                        (                                                                                    3                              4                                                        -                            η                                                    )                                                ⁢                                                  λ                          ′                                                                                      )                                                  ]                            +                              exp                [                                                                                                    -                        2                                            ⁢                      π                      ⁢                                                                                          ⁢                      i                                                              λ                      ′                                                        ⁢                                      (                                                                                                                        r                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                2                                                                                                      λ                                                          λ                              ′                                                                                ⁢                          f                                                                    -                                                                        (                                                                                    3                              4                                                        -                            η                                                    )                                                ⁢                                                  λ                          ′                                                                                      )                                                  ]                                      ]                                              [                  Formula          ⁢                                          ⁢          5                ]            
Here, the first term of Formula 5 is a transmitted wave (zeroth-order diffracted wave), the second term is the spherical wave (primary diffracted wave) that diverges from the point light source, and the third term is the spherical wave (negative primary diffracted wave) that converges and forms a point image. This is nothing but a reproduction process of holography, and denoting a distance from the zone plate to the point image by f′, f′ just changes to f′=λ/λ′f and the initial phase just changes to ¾−η. The initial phase is a term that disappears by being cancelled out when the intensity distribution as an image is discussed, and the image formation distance f′ contains a meaning of variation of magnification accompanying the change of wavelength.
Although a diffraction effect expressed by Formula 5 covers as large as the zeroth order term and ±first order terms at the most, when the effect is accompanied with higher order diffraction effects as in the case of a binarized zone plate, real images and virtual images by the higher order diffracted waves are also formed. However, one into which the intensity is concentrated most among the diffracted waves are ±first order diffracted waves, and their distances from the zone plate are expressed by Formula 6 as the principal focal distance f1.
                              f          1                =                              r            1            2                                λ            ′                                              [                  Formula          ⁢                                          ⁢          6                ]            
Here, r1 is a radius at which Formula 3 becomes ½ for the first time, namely, a radius r1=√(λf) of an innermost annular zone when binarization is done, and λ′ is a wavelength of the wave that irradiates the zone plate. When λ′=λ, f1 coincides with a distance f′ between the zone plate and the point light source.
Next, consider a case where the spherical wave Φs expressed by Formula 2 is entered into the zone plate with an amplitude transmissivity Ψt expressed by Formula 4. Then, the wave immediately after its penetration of the zone plate is expressed by Formula 7. Here, because of simplicity, kt=1 is assumed and the same wavelength λ as that at the time of interference fringe recording is used.
                                          Φ            re                    ⁡                      (                          x              ,              y                        )                          =                                            1              2                        ⁢                          exp              ⁡                              [                                                                            2                      ⁢                      π                      ⁢                                                                                          ⁢                      i                                        λ                                    ⁢                                                                                    r                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    2                                        f                                                  ]                                              +                                    1              2                        ⁡                          [                                                exp                  ⁡                                      [                                                                                            2                          ⁢                          π                          ⁢                                                                                                          ⁢                          i                                                λ                                            ⁢                                              (                                                                                                                                            r                                ⁡                                                                  (                                                                      x                                    ,                                    y                                                                    )                                                                                            2                                                                                      f                              2                                                                                -                                                                                    3                              ⁢                              λ                                                        4                                                                          )                                                              ]                                                  +                                  exp                  ⁡                                      [                                                                                            2                          ⁢                          π                          ⁢                                                                                                          ⁢                          i                                                λ                                            ⁢                                                                        3                          ⁢                          λ                                                4                                                              ]                                                              ]                                                          [                  Formula          ⁢                                          ⁢          7                ]            
A first term of Formula 7 expresses a transmitted wave (zeroth-order diffracted wave), a second term expresses the spherical wave whose focus distance is one half and whose degree of divergence is large (primary diffracted wave), and a third term expresses the plane wave (− primary diffracted wave). That is, the plane wave is reproduced even when the spherical wave is made to enter. This is one manifestation of the lens effect of the zone plate. The zone plate is capable of generating the spherical wave and the plane wave, respectively, when either of the spherical wave and the plane wave is made to enter.
<Brightness of Zone Plate>
Since the zone plate behaves as a lens, even when it is used in the optical system by specifying any area on the plate as the opening, it performs imaging performance as described above. However, since the interference phenomenon is used fundamentally, when the zone plate comes to a state where the innermost circular opening in the central part is blocked partially, it is known that a brightness of the image decreases rapidly.
This originates in that the opening area of the central part that contributes to the brightness most decreases in the positive zone plate and that an effect of mutually strengthening by an interference of the waves from the positions of symmetry across the center is lost. Therefore, when using the zone plate as a fast lens, it is necessary to consider not only a size of the opening of the optical system but also a fact that the center of the zone plate is included within the opening.
<Spiral Wave>
In a coherent optical system, a phase of the light wave that propagates is uniquely determined. A plane where its phase is equal is called a wave front, and from the shape of the wave front, classification of the wave fronts is carried out into the plane wave, the spherical wave, etc. This will be described in detail later using a formula. For example, a wave whose equiphase surface has a spiral shape centering on a certain axis (generally, parallel to the optical axis) shall be called the spiral wave in this application. FIG. 3 schematically shows a spiral wave 88 that is classified into the plane wave. In this wave, a singular point of the phase exists on a spiral axis, and it is impossible to define a phase on this axis.
This spiral wave is called a Laguerre-Gaussian beam or an optical spiral (optical vortex) in the optics, is the light wave propagating with an orbital angular momentum preserved, and can exert a force on the equiphase surface (wave front) perpendicularly thereto. Therefore, it becomes able to give momentum to an irradiation object, and is put in practical use, for example, as a manipulation technology such as an optical tweezer for operating particles about a size of a cell, and as laser processing and a super-resolution spectromicroscopy.
Furthermore, since multiple orbital angular momentums can be made to exist in a portion of the spiral axis that is a phase singularity point (as a topology charge, a strength of winding of the spiral can be selected), new technical deployments are expected in physical property analyses and structural analysises such as analyses of a magnetic state and a stereoscopic model of atomic arrangement in the fields of quantum communications and in the X-ray.
Also in the electron ray, in the 1970s when crystal observation by multiple-wave image formation has begun to be carried out, a discontinuous grating image was recorded in the high resolution image as an image of an edge dislocation in a crystal lattice, and existence of an electron wave having the spiral form was known. Moreover, in the reflection type electron ray holography, a phase image whose equiphase surface just forms the spiral shape was reproduced in a portion where the spiral dislocation reached to a surface (Nonpatent Literature 2). However, there is no case where an electron spiral wave is positively used as a probe.
After the spiral wave (Laguerre-Gaussian beam) attracted attention as a new probe in the field of the light wave (Nonpatent Literature 1), trials of generating the spiral waves positively have been performed also with electron rays. An electron ray is transmitted through a graphite membrane whose thickness varied spirally (Nonpatent Literature 3), a spectroscopic method using a diffraction grating containing an edge dislocation (forked diffraction grating) (Nonpatent Literature 4) has been tried, and formation of the electron spiral wave has been confirmed because a diffraction point has a ring-like intensity distribution.
In this spectroscopic method, an end part where the grating fringe is broken to become discontinuous is considered as an edge dislocation. It is possible to draw multiple grating fringes from one end part and, for example, in the case where two fringes are drawn, it is possible to generate the spiral wave whose phase varies by 4π (two wavelengths) when it goes around the spiral axis. This is called a secondary spiral and it is called a second-order edge dislocation. A spiral degree and an edge dislocation degree coincide with each other. Generation of a 25th-order spiral wave has been experimentally checked using a 25th-order edge dislocation.
Since the electron spiral wave makes the electron ray propagates with the orbital angular momentum preserved, it is expected that the electron spiral wave produces application fields as a probe of the electron ray that does not exist thus far. For example, they include sensitivity improvement and a three-dimensional state measurement in magnetization measurements, high-contrast and high-resolution observations of a protein molecule and a carbohydrate chain, etc.
Particularly, in magnetization observation, although the electron ray has a fundamentally fault that the electron ray does not have sensitivity to magnetization in parallel to its propagation direction, there is a possibility that the electron spiral wave can observe the magnetization in the propagation direction of the electron ray. Because of this reason, the electron spiral wave is beginning to be brought into the limelight as a probe of a next-generation electron ray apparatus.